To prove that something always happens, assume the hypothesis for an unnamed case and reason to the conclusion using only what the assumption gives. Because the case was never named, the argument covers all of them at once. To disprove such a claim you need only one case where it fails.

Checking examples is like testing a bridge by driving a few cars over it: encouraging, but no guarantee. A direct proof is the engineer’s calculation, which covers every vehicle within the load limit. One collapsing car, meanwhile, ends the discussion.

Almost every claim in this course has the form of a universal implication. The proof begins by taking an arbitrary object satisfying the hypothesis and ends at the conclusion, using nothing about the object beyond the hypothesis itself.

• Assume $P(n)$ for an arbitrary $n$, then derive $Q(n)$. The claim follows for every $n$.
• The object must stay arbitrary. Using anything particular about it proves only that particular case.
• To disprove the claim, exhibit a single $n$ with true and false.

Write the assumption as an equation, square it, and rearrange until the definition of evenness is visible again. Nothing about the particular number is used beyond that first equation.

Take an arbitrary even number; nothing else will be assumed about it.

This is what evenness means, written as an equation we can work with.

Square both sides of that equation.

Expand the square on the right.

Pull out a factor of two, which is the form the definition asks for.

$n^2$ is twice an integer, which is exactly what it means to be even.

Let $n$ be even, so $n = 2k$.

Nothing here says which even number $n$ is, which is exactly why the conclusion will hold for all of them.

$n^2 + n + 41$ is prime for $n = 0,1,2,\dots,39$, but not for .

Forty agreeing examples, and the claim still fails. Only a proof rules out the case waiting further along.

For "if $n$ is even then $n > 5$", take $n = 2$.

$2$ is even and is not greater than five, so the hypothesis holds while the conclusion fails.

The word "even" becomes an equation, and from there ordinary algebra can proceed.

When the hypothesis offers alternatives, argue each one, and check that together they cover everything assumed.

• A direct proof assumes the hypothesis for an arbitrary object and derives the conclusion.
• The object must stay arbitrary, and every definition used should be written out.
• A proof by cases is a direct proof whose cases must cover the hypothesis.
• One counterexample disproves a universal claim; examples never prove one.

Great work! You now know how to:

• Open a direct proof with an arbitrary object and the definitions written out
• Split a hypothesis into cases that cover it
• Disprove a universal claim with a single counterexample